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Reciprocal Identities: Uses & Applications

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  • 0:03 The Reciprocal Identities
  • 1:04 Uses
  • 2:21 Example 1
  • 3:08 Example 2
  • 3:53 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Trigonometry is full of identities, a set of which are called the reciprocal identities. Watch this video lesson to learn what they are and how you can use them to help you solve problems.

The Reciprocal Identities

In this video lesson, we talk about the reciprocal identities of trigonometry. What are they? They are the definitions of our trig functions in terms of another trig function. They tell us how the trig functions are related to each other. They also tell us which trig functions are reciprocals of each other.

Remember in math a reciprocal of a number is 1 divided by that number? For example, the reciprocal of 5 is 1/5. Well, in trigonometry, the reciprocal of a trig function is 1 divided by another trig function. What are they? Here are six of them:

This trig function Is equal to
sin (theta) 1/csc (theta)
cos (theta) 1/sec (theta)
tan (theta) 1/cot (theta)
cot (theta) 1/tan (theta)
sec (theta) 1/cos (theta)
csc (theta) 1/sin (theta)

Look carefully and you will spot corresponding pairs. For example, because the sine function is equal to 1 over the cosecant function, we also have the cosecant function is equal to 1 over the sine function. Notice how the two corresponding trig functions have simply switched places? We have six statements, so you will see three corresponding pairs. Can you spot them?

Uses

Now we know what the reciprocal identities are. So, what can we do with them? We can use these identities, or true statements, to help us simplify trig problems. We do this by substituting our definitions into the problem where we can to help us simplify and then solve the problem.

For example, the problem sin^2 (pi / 2) * csc (pi / 2) might seem difficult at first. However, we see that we have a sine function, as well as a cosecant function. We know that they are reciprocal functions. We can actually substitute 1 over sine in for the cosecant.

After we do that, our function becomes very simple. With the reciprocal, one of our sine functions can be canceled and we are left with just a single sine of pi/2. pi/2 is a radian measure. We can use either our unit circle to find our answer or we can input this into a calculator, making sure our calculator is set to radians. Doing this we see that we get a nice answer of 1:

reciprocal identities

A good rule of thumb to follow to make your problem solving easier is to rewrite your functions into sine, cosine, and tangent. So, if you see two functions that are reciprocals of each other, rewrite the one that isn't the sine, cosine, or tangent function.

Example 1

Let's look at a couple more examples to see how we can use our reciprocal identities. Look at this problem: Simplify cos (theta) * sec (theta) * tan (theta) * cot (theta).

At first look, we might wonder how we could possibly simplify this? We have four functions all being multiplied together. How can we possibly cancel anything out?

Not to worry, our reciprocal identities are here to help us out. We have a cosine and a secant. They are reciprocals. So, we can write our secant as 1 over cosine; the same with tangent and cotangent. We can write our cotangent as 1 over tangent.

What happens after we do that? We can start canceling out terms. What are we left with? Why, a big, simple 1! Yay! That was very cool!

reciprocal identities

Example 2

Let's look at one more:

reciprocal identities

How can we simplify this? We see that we have a sine function, a secant function, and a cosine function. Which ones are reciprocals of each other?

The secant and cosine. We can rewrite the secant function as 1 over cosine. But wait, our secant is already in the denominator. What do we do?

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